The author's prior development of a general abstract representation for the design sensitivities of Green's functional for linear structural systems is extended to the case where the structural stiffness vanishes at an internal location. This situation often occurs in the optimal design of structures. Most optimality criteria require that optimally designed beams be statically determinate. For clamped-pinned beams, for example, this is possible only if the flexural stiffness vanishes at some intermediate location. The Green's function for such structures depends upon the stiffness and the location where it vanishes. A precise representation for Green's function's sensitivity to the location of vanishing stiffness is presented for beams and axisymmetric plates.